Finite Group DatabasebetaFinite Group Databasebeta| order | 1176 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×ΓL(1,𝔽8)⋊Dic7 |
| unambiguous semidirect product | (C2×ΓL(1,𝔽8))⋊Dic7 |
| derived, Sylow | C72 ⋊ C2×C3×C4 |
| Hall | C2×C7⋊Dic7 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | ΓL(1,𝔽8)⋊D7 |
| commutator, hypocenter | C72 | C2×C3×C4 |
| Frattini | C2 | C2×ΓL(1,𝔽8)⋊D7 |
| Fitting, socle | C22×C72 | C2×C3 |
| prime radical | C22×C7×ΓL(1,𝔽8) | C2 |
| cosocle | C2×C72 | C22×C3 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C22 | |
80 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
88 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1176, 116);